Lebesgue decomposition in action via semidefinite relaxations

TL;DR

This paper introduces a numerical scheme using semidefinite relaxations to decompose a measure into absolutely continuous and singular parts based solely on finite moments, without prior support knowledge.

Contribution

It provides a novel method to compute Lebesgue decompositions from moments using semidefinite programming, with convergence guarantees.

Findings

Convergent sequences of moments for the decomposed measures.

No prior support information needed for the decomposition.

Applicable to measures with densities in $L_\infty(\lambda)$.

Abstract

Given all (finite) moments of two measures $\mu$ and $\lambda$ on $\R^n$, we provide a numerical scheme to obtain the Lebesgue decomposition $\mu=\nu+\psi$ with $\nu\ll\lambda$ and $\psi\perp\lambda$. When$\nu$ has a density in $L\_\infty(\lambda)$ then we obtain two sequences of finite moments vectorsof increasing size (the number of moments) which converge to the moments of $\nu$ and $\psi$ respectively, as the number of moments increases. Importantly, {\it no} \`a priori knowledge on the supports of $\mu, \nu$ and $\psi$ is required.